We can define \(\gamma ^{35}=i\gamma ^{3}\gamma ^{5}=\tau ^{3}\otimes i\sigma ^{0}\)  which commutes with all \(\gamma ^{\mu }\)  but anticommutes with \(\gamma ^{3}\)  and \(\gamma ^{5}\) , or explicitly, \([\gamma ^{35},\gamma ^{\mu }]=0\) , \(\lbrace \gamma ^{35},\gamma ^{3}\rbrace =0\) , \(\lbrace \gamma ^{35},\gamma ^{5}\rbrace =0\) . It's
easily check that \([\gamma ^{35},\mathcal {H}]=0\) . The Hamiotonian possesses
a symmetry implemented by \(C\mathcal {H}C^{-1}=-\mathcal {H}\) , where \(C\)  is
expressed as either \(C=\gamma ^{0}\)  or \(C=\gamma ^{0}\gamma ^{35}\) . This
symmetry is conventionally called chiral symmetry or sublattice symmetry on
bipartite graphene lattice. For generality, we introduce an arbitrary number
of \(N_{f}\)  fermion flavors of four-component Dirac fermions. The fermion
field carries a flavor index, \(\Psi =\Psi _{i}\)  with \(i=\lbrace 1,2,...N_{f}\rbrace \) , \(N_{f}=2\)  case corresponds to spin-1/2 fermions on the honeycomb lattice[1]}, [2]}.
